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On the development of symmetry-preserving finite element schemes for ordinary differential equations
A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems
School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia |
Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.
References:
[1] |
M. R. Allshouse and T. Peacock, Lagrangian based methods for coherent structure detection, Chaos, 25 (2015), 13pp.
doi: 10.1063/1.4922968. |
[2] |
S. Balasuriya, N. T. Ouellette and I. I. Rypina,
Generalized Lagrangian coherent structures, Phys. D, 372 (2018), 31-51.
doi: 10.1016/j.physd.2018.01.011. |
[3] |
R. Banisch and P. Koltai, Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets, Chaos, 27 (2017), 16pp.
doi: 10.1063/1.4971788. |
[4] |
M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33pp.
doi: 10.1063/1.4772195. |
[5] |
A. J. Charlton, A. O'Neill, W. A. Lahoz and P. Berrisford,
The splitting of the stratospheric polar vortex in the Southern Hemisphere, September 2002: Dynamical evolution, J. Atmospheric Sci., 62 (2005), 590-602.
doi: 10.1175/JAS-3318.1. |
[6] |
D. Dee, S. Uppala, A. Simmons, P. Berrisford and P. Poli, et al., The ERA-Interim reanalysis: Configuration and performance of the data assimilation system, Quarterly J. Roy. Meteorological Soc., 137 (2011), 553–597.
doi: 10.1002/qj.828. |
[7] |
M. Dellnitz, G. Froyland, C. Horenkamp and K. Padberg,
On the approximation of transport phenomena—A dynamical systems approach, GAMM-Mitt., 32 (2009), 47-60.
doi: 10.1002/gamm.200910004. |
[8] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, (2001), 145–174,805–807. |
[9] |
M. Dellnitz and O. Junge,
On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[10] |
P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Lecture Notes in Computational Science and Engineering, 4, Springer, Berlin, Heidelberg, 1999, 98–115.
doi: 10.1007/978-3-642-58360-5_5. |
[11] |
G. Froyland, T. Hüls, G. P. Morriss and T. M. Watson,
Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.
doi: 10.1016/j.physd.2012.12.005. |
[12] |
G. Froyland, S. Lloyd and A. Quas,
Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[13] |
G. Froyland, S. Lloyd and A. Quas,
A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.
doi: 10.3934/dcds.2013.33.3835. |
[14] |
G. Froyland, S. Lloyd and N. Santitissadeekorn,
Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[15] |
G. Froyland and K. Padberg,
Almost-invariant sets and invariant manifolds—Connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[16] |
G. Froyland, K. Padberg, M. H. England and A. M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007).
doi: 10.1103/PhysRevLett.98.224503. |
[17] |
G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat., Springer, New York, 70 (2014), 171–216.
doi: 10.1007/978-1-4939-0419-8_9. |
[18] |
G. Froyland, C. P. Rock and K. Sakellariou,
Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 81-107.
doi: 10.1016/j.cnsns.2019.04.012. |
[19] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 10pp.
doi: 10.1063/1.3502450. |
[20] |
F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007).
doi: 10.1103/PhysRevLett.99.130601. |
[21] |
C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., Amer. Math. Soc., Providence, RI, 709 (2018), 31–52.
doi: 10.1090/conm/709/14290. |
[22] |
C. González-Tokman and A. Quas,
A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.
doi: 10.1017/etds.2012.189. |
[23] |
C. González-Tokman and A. Quas,
A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.
doi: 10.3934/jmd.2015.9.237. |
[24] |
G. Haller, Lagrangian coherent structures, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 47 (2015), 137–162.
doi: 10.1146/annurev-fluid-010313-141322. |
[25] |
G. Haller, D. Karrasch and F. Kogelbauer,
Material barriers to diffusive and stochastic transport, Proc. Natl. Acad. Sci. USA, 115 (2018), 9074-9079.
doi: 10.1073/pnas.1720177115. |
[26] |
B. Joseph and B. Legras,
Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex, J. Atmospheric Sci., 59 (2002), 1198-1212.
doi: 10.1175/1520-0469(2002)059<1198:RBKBSA>2.0.CO; 2. |
[27] |
S. Klus, P. Koltai and C. Schütte,
On the numerical approximation of the Perron-Frobenius and Koopman operator, J. Comput. Dyn., 3 (2016), 51-79.
doi: 10.3934/jcd.2016003. |
[28] |
P. Koltai and D. R. M. Renger,
From large deviations to semidistances of transport and mixing: Coherence analysis for finite Lagrangian data, J. Nonlinear Sci., 28 (2018), 1915-1957.
doi: 10.1007/s00332-018-9471-0. |
[29] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[30] |
F. Lekien and S. D. Ross, The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds, Chaos, 20 (2010), 20pp.
doi: 10.1063/1.3278516. |
[31] |
B. A. Mosovsky and J. D. Meiss,
Transport in transitory dynamical systems, SIAM J. Appl. Dyn. Syst., 10 (2011), 35-65.
doi: 10.1137/100794110. |
[32] |
M. Ndour and K. Padberg-Gehle, Predicting bifurcations of almost-invariant patterns: A set-oriented approach, preprint, arXiv: 2001.01099. Google Scholar |
[33] |
P. Newman and E. Nash,
The unusual Southern Hemisphere stratosphere winter of 2002, J. Atmospheric Sci., 62 (2005), 614-628.
doi: 10.1175/JAS-3323.1. |
[34] |
F. Noethen,
A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.
doi: 10.1016/j.physd.2019.02.012. |
[35] |
A. O'Neill, C. L. Oatley, A. J. Charlton–Perez, D. M. Mitchell and T. Jung,
Vortex splitting on a planetary scale in the stratosphere by cyclogenesis on a subplanetary scale in the troposphere, Quarterly J. Roy. Meteorological Soc., 143 (2017), 691-705.
doi: 10.1002/qj.2957. |
[36] |
Y. J. Orsolini, C. E. Randall, G. L. Manney and D. R. Allen,
An observational study of the final breakdown of the Southern Hemisphere stratospheric vortex in 2002, J. Atmospheric Sci., 62 (2005), 735-747.
doi: 10.1175/JAS-3315.1. |
[37] |
V. I. Oseledec,
A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.
|
[38] |
K. Padberg-Gehle, S. Reuther, S. Praetorius and A. Voigt, Transfer operator-based extraction of coherent features on surfaces, in Topological Methods in Data Analysis and Visualization. IV, Math. Vis., Springer, Cham, (2017), 283–297.
doi: 10.1007/978-3-319-44684-4_17. |
[39] |
M. S. Raghunathan,
A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.
doi: 10.1007/BF02760464. |
[40] |
S. C. Shadden, F. Lekien and J. E. Marsden,
Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Phys. D, 212 (2005), 271-304.
doi: 10.1016/j.physd.2005.10.007. |
[41] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, 1960. |
[42] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
show all references
References:
[1] |
M. R. Allshouse and T. Peacock, Lagrangian based methods for coherent structure detection, Chaos, 25 (2015), 13pp.
doi: 10.1063/1.4922968. |
[2] |
S. Balasuriya, N. T. Ouellette and I. I. Rypina,
Generalized Lagrangian coherent structures, Phys. D, 372 (2018), 31-51.
doi: 10.1016/j.physd.2018.01.011. |
[3] |
R. Banisch and P. Koltai, Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets, Chaos, 27 (2017), 16pp.
doi: 10.1063/1.4971788. |
[4] |
M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos, 22 (2012), 33pp.
doi: 10.1063/1.4772195. |
[5] |
A. J. Charlton, A. O'Neill, W. A. Lahoz and P. Berrisford,
The splitting of the stratospheric polar vortex in the Southern Hemisphere, September 2002: Dynamical evolution, J. Atmospheric Sci., 62 (2005), 590-602.
doi: 10.1175/JAS-3318.1. |
[6] |
D. Dee, S. Uppala, A. Simmons, P. Berrisford and P. Poli, et al., The ERA-Interim reanalysis: Configuration and performance of the data assimilation system, Quarterly J. Roy. Meteorological Soc., 137 (2011), 553–597.
doi: 10.1002/qj.828. |
[7] |
M. Dellnitz, G. Froyland, C. Horenkamp and K. Padberg,
On the approximation of transport phenomena—A dynamical systems approach, GAMM-Mitt., 32 (2009), 47-60.
doi: 10.1002/gamm.200910004. |
[8] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO-set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, (2001), 145–174,805–807. |
[9] |
M. Dellnitz and O. Junge,
On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515.
doi: 10.1137/S0036142996313002. |
[10] |
P. Deuflhard, M. Dellnitz, O. Junge and C. Schütte, Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Lecture Notes in Computational Science and Engineering, 4, Springer, Berlin, Heidelberg, 1999, 98–115.
doi: 10.1007/978-3-642-58360-5_5. |
[11] |
G. Froyland, T. Hüls, G. P. Morriss and T. M. Watson,
Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Phys. D, 247 (2013), 18-39.
doi: 10.1016/j.physd.2012.12.005. |
[12] |
G. Froyland, S. Lloyd and A. Quas,
Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[13] |
G. Froyland, S. Lloyd and A. Quas,
A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.
doi: 10.3934/dcds.2013.33.3835. |
[14] |
G. Froyland, S. Lloyd and N. Santitissadeekorn,
Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[15] |
G. Froyland and K. Padberg,
Almost-invariant sets and invariant manifolds—Connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[16] |
G. Froyland, K. Padberg, M. H. England and A. M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007).
doi: 10.1103/PhysRevLett.98.224503. |
[17] |
G. Froyland and K. Padberg-Gehle, Almost-invariant and finite-time coherent sets: Directionality, duration, and diffusion, in Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proc. Math. Stat., Springer, New York, 70 (2014), 171–216.
doi: 10.1007/978-1-4939-0419-8_9. |
[18] |
G. Froyland, C. P. Rock and K. Sakellariou,
Sparse eigenbasis approximation: Multiple feature extraction across spatiotemporal scales with application to coherent set identification, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 81-107.
doi: 10.1016/j.cnsns.2019.04.012. |
[19] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 10pp.
doi: 10.1063/1.3502450. |
[20] |
F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi and A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007).
doi: 10.1103/PhysRevLett.99.130601. |
[21] |
C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, in Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, Contemp. Math., Amer. Math. Soc., Providence, RI, 709 (2018), 31–52.
doi: 10.1090/conm/709/14290. |
[22] |
C. González-Tokman and A. Quas,
A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.
doi: 10.1017/etds.2012.189. |
[23] |
C. González-Tokman and A. Quas,
A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.
doi: 10.3934/jmd.2015.9.237. |
[24] |
G. Haller, Lagrangian coherent structures, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 47 (2015), 137–162.
doi: 10.1146/annurev-fluid-010313-141322. |
[25] |
G. Haller, D. Karrasch and F. Kogelbauer,
Material barriers to diffusive and stochastic transport, Proc. Natl. Acad. Sci. USA, 115 (2018), 9074-9079.
doi: 10.1073/pnas.1720177115. |
[26] |
B. Joseph and B. Legras,
Relation between kinematic boundaries, stirring, and barriers for the Antarctic polar vortex, J. Atmospheric Sci., 59 (2002), 1198-1212.
doi: 10.1175/1520-0469(2002)059<1198:RBKBSA>2.0.CO; 2. |
[27] |
S. Klus, P. Koltai and C. Schütte,
On the numerical approximation of the Perron-Frobenius and Koopman operator, J. Comput. Dyn., 3 (2016), 51-79.
doi: 10.3934/jcd.2016003. |
[28] |
P. Koltai and D. R. M. Renger,
From large deviations to semidistances of transport and mixing: Coherence analysis for finite Lagrangian data, J. Nonlinear Sci., 28 (2018), 1915-1957.
doi: 10.1007/s00332-018-9471-0. |
[29] |
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Applied Mathematical Sciences, 97, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[30] |
F. Lekien and S. D. Ross, The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds, Chaos, 20 (2010), 20pp.
doi: 10.1063/1.3278516. |
[31] |
B. A. Mosovsky and J. D. Meiss,
Transport in transitory dynamical systems, SIAM J. Appl. Dyn. Syst., 10 (2011), 35-65.
doi: 10.1137/100794110. |
[32] |
M. Ndour and K. Padberg-Gehle, Predicting bifurcations of almost-invariant patterns: A set-oriented approach, preprint, arXiv: 2001.01099. Google Scholar |
[33] |
P. Newman and E. Nash,
The unusual Southern Hemisphere stratosphere winter of 2002, J. Atmospheric Sci., 62 (2005), 614-628.
doi: 10.1175/JAS-3323.1. |
[34] |
F. Noethen,
A projector-based convergence proof of the Ginelli algorithm for covariant Lyapunov vectors, Phys. D, 396 (2019), 18-34.
doi: 10.1016/j.physd.2019.02.012. |
[35] |
A. O'Neill, C. L. Oatley, A. J. Charlton–Perez, D. M. Mitchell and T. Jung,
Vortex splitting on a planetary scale in the stratosphere by cyclogenesis on a subplanetary scale in the troposphere, Quarterly J. Roy. Meteorological Soc., 143 (2017), 691-705.
doi: 10.1002/qj.2957. |
[36] |
Y. J. Orsolini, C. E. Randall, G. L. Manney and D. R. Allen,
An observational study of the final breakdown of the Southern Hemisphere stratospheric vortex in 2002, J. Atmospheric Sci., 62 (2005), 735-747.
doi: 10.1175/JAS-3315.1. |
[37] |
V. I. Oseledec,
A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.
|
[38] |
K. Padberg-Gehle, S. Reuther, S. Praetorius and A. Voigt, Transfer operator-based extraction of coherent features on surfaces, in Topological Methods in Data Analysis and Visualization. IV, Math. Vis., Springer, Cham, (2017), 283–297.
doi: 10.1007/978-3-319-44684-4_17. |
[39] |
M. S. Raghunathan,
A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.
doi: 10.1007/BF02760464. |
[40] |
S. C. Shadden, F. Lekien and J. E. Marsden,
Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Phys. D, 212 (2005), 271-304.
doi: 10.1016/j.physd.2005.10.007. |
[41] |
S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London, 1960. |
[42] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
























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